The generator matrix

 1  1  X  1  1
 0 X^3+X^2 X^3+X^2  0  0
 0  0 X^3  0  0
 0  0  0 X^3  0

generates a code of length 5 over Z2[X]/(X^4) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+7x^2+40x^4+160x^5+40x^6+7x^8+1x^10

The gray image is a linear code over GF(2) with n=40, k=8 and d=8.
This code was found by Heurico 1.16 in -3.24e-008 seconds.